I had this very same issue when I got my K3, well in fact it was an issue waiting to be resolved before that but budgets needed to be funded and that takes time :-( There's lot's of good advice on here but I thought I would just add my list as I have been very happy with what I now have for over a year.
Having had like you, a very very old PC and mine had a pathetic amount of RAM etc I decided I wasn't going to struggle anymore with my PC being rendered a useless piece of metal which just made noises and wasted my time.
What I have now does a very nice job thank you; I'm sure people might say you don't need all of that and they might be right but hey I am very very happy with what I have, it works and works very well, I don't spend my time fighting with technology when I just want to get a job done so this is what I now have:
Mac book pro 15inch Retina (late 2013) - This monitor can be calibrated by third party monitor calibration equipment - worth doing.
2.3 GHz Intel Core i7
16 GB 1600 MHz DDR3
NVIDIA GeForce GT 750M 2048 MB
500 Gigabyte Solid State drive
External 3 Terabyte storage + another 1 Terabyte external drive which are synched
24'' external monitor - nothing special but will soon be replaced by a monitor that can be calibrated, those budgets again
This just sings along without a worry, I thought about an apple desktop but negations with my wife and the fact we don't have much space forced the laptop route but it still works well. The SSD memory is excellent, quick, robust and quiet :-) oh I do like quiet.
My only issue is that the internal 500 Gigabytes of SSD is too small for my photo collection, which in a way might be a good thing and not an issue if I wasn't so lazy, it means you need some discipline in managing your photo collections and having multiple copies of them synched for security.
As for using the cloud for my photo collection well that's just a joke the cost and performance issues make it a non starter.
Software:
Since the sad demise of Aperture's future I have been a refugee in search of a new workflow home, I shoot in RAW or RAW + Jpeg; I have procrastinated but it comes down to Lightroom or capture One or maybe the new Affinity Photo by Serif (Beta) (Mac only) .. did I say I procrastinated? I would advise getting the 30 day trials and have a proper play.
Good luck